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From Harald Helfgott's proof of Goldbach's weak conjecture,1 we know that every odd number $> 7$ is the sum of three odd primes. If $n$ is such an odd number, say that two sums that yield $n$ are distinct if they are distinct as sorted lists. So $n=13$ has just two distinct sums, $(3,3,7)$ and $(3,5,5)$.

Q1. Are $9=3+3+3$ and $11 = 3+3+5$ the only two odd numbers $> 7$ that have unique sums of three odd primes?

Call a sum of three odd primes as a $3$-sum for conciseness. Here are the first few odd numbers and their $3$-sum lists: $$ \left| \begin{array}{ccc} n & 3{-}sums & \#3{-}sums\\ \hline 9 & (3,3,3) & 1\\ 11 & (3,3,5) & 1\\ 13 & (3, 3, 7),\; (3, 5, 5) & 2\\ 15 & (3, 5, 7),\; (5, 5, 5) & 2\\ 17 & (3, 3, 11),\; (3, 7, 7),\; (5, 5, 7) & 3 \end{array} \right| $$

Here is a plot of the number of $3$-sums for odds up to about $1001$:

          Goldbachs_1000

Q2. What explains the separation into two distinguishable series?

So for $n=975$, there are $701$ $3$-sums:

\begin{eqnarray} &(3, 5, 967)\\ &(3, 19, 953)\\ &(3, 31, 941)\\ &\cdots\\ &(313, 313, 349)\\ &(313, 331, 331) \end{eqnarray}

But for $n=977$, there are $1106$ $3$-sums. Here is a bit more data: $$ \left| \begin{array}{cc} n & \#3{-}sums \\ \hline 975 & 701 \\ 977 & 1106 \\ 979 & 1010 \\ 981 & 758 \\ 983 & 1113 \\ 985 & 977 \\ 987 & 756 \\ 989 & 1110 \\ 991 & 1037 \\ 993 & 791 \\ 995 & 1056 \\ 997 & 1096 \\ 999 & 770 \\ 1001 & 1095 \\ \end{array} \right| $$

1MO: Proof of the weak Goldbach Conjecture .

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    $\begingroup$ Did you think about mod 3 considerations? Gerhard "Can You Finish The Thought?" Paseman, 2019.06.18. $\endgroup$
    – Gerhard Paseman
    Jun 19, 2019 at 0:06
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    $\begingroup$ Presumably Helfgott and Platt's computations determined that the answer to Q1 is "yes" among the first 10-to-the-umpteen integers, and Helfgott's explicit bounds for larger integers show that the answer to Q1 is "yes" for the rest. $\endgroup$
    – Greg Martin
    Jun 19, 2019 at 0:32
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    $\begingroup$ Have you considered the integers 'many' 3-sums of which are of the form $(p, p, q)$ with $p\neq q$? $\endgroup$
    – Sylvain JULIEN
    Jun 19, 2019 at 10:09
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    $\begingroup$ Are you familiar, Joseph, with "Goldbach's Comet"? $\endgroup$
    – Gerry Myerson
    Jun 19, 2019 at 13:05
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    $\begingroup$ @GerryMyerson: I did not know about Goldbach's comet. What a great name! Clearly my plot is a (monochromatic) version of the comet for triples rather than pairs. I appreciate learning of this---Thanks! $\endgroup$
    – Joseph O'Rourke
    Jun 19, 2019 at 14:49

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